Continuity mistake: When Max holds the cables and sings happy birthday to himself, the position of his left-hand changes, from more distant to the ring of the connector in the view from underneath, and closer to it in the other angle. (00:36:10)
Continuity mistake: When Max is talking to his boss and fantasizes to assault him, there's a woman walking past them reading some documents, and she keeps coming from the corridor and passing them by - that's without considering Max's fantasy, just shots of the 'real' events. (00:27:50)
Deliberate mistake: When Spidey jumps on the hood of the truck, Rhino squirts him using the wipers. Funny, but he can do that only because the detergent sprays forward instead of the angle actual windscreen washers are pointed at. (00:12:50)
Other mistake: We see a good view of Spiderman's suit during his acrobatics in the air at the beginning of the movie; it's skin-tight and has no compartment or belt. However, when Gwen calls, Spidey reaches behind his back and pulls his phone out. (00:11:30)
Other mistake: This movie is set in 2014 (obituary of Norman Osborn) and Peter's parents disappeared in 2002 (articles printed by Peter, FBI files), but in the first movie the young version of the protagonist was credited as " Peter Parker (age 4). " Peter Parker is not supposed to be a 16 years old boy in this movie which opens with his graduation.
Other mistake: The first shot of the movie is an ominous zoom-out from Richard Parker's watch, a top quality Swiss watch...although, it shows that it's 5:42, and the timestamp in the security cameras is 21:05. (00:00:55 - 00:01:25)
Continuity mistake: On the plane, when Peter's father holds the mother's hand she has a small red notebook next to her. When the angle changes it has swapped to a huge red book. (00:03:15)
Continuity mistake: Harry and Electro are in the office and confront the new CEO. Electro shocks him, and he falls to the floor. The first shot of him is clearly not the same actor, because the man on the floor has a full head of hair. The second shot it's the original CEO, who is mostly bald. (01:40:55)
Other mistake: After Electro gets the two cables on his chest, his clothes do not burn and have no marks. (00:58:05)
Suggested correction: Actually they do burn, since they emit smoke very noticeably. They don't show marks, true. One could argue that Electro does pretty weird things with his clothes in general in this movie, since he has underpants resistant to anything and his clothes of choice stay with him even when he switches from pure energy to solid form, but make of that argument what you will.
Continuity mistake: When the kid slowly hobbles towards Rhino somehow outsprinting all the policemen, seen from behind he is walking in the shade for a large tract, but the street is sunlit in the other views. In general throughout the scene the police cars are now in the shade, now in the sun. (02:10:30)
Character mistake: During the scene at the office of the University of Oxford, the Union Jack is hanged the opposite way from the proper position. (01:23:10)